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Question

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 6 to 6 in increments of 1. Three triangles are formed by joining three plotted points each. The coordinates of the plotted points for the first triangle upper A upper B upper C joined by solid lines are upper A is left parenthesis negative 4 comma 5 right parenthesis, upper B is left parenthesis negative 1 comma 3 right parenthesis, and upper C is left parenthesis negative 3 comma 1 right parenthesis. The coordinates for the second triangle upper A prime upper B prime upper C prime joined by dotted lines are as follows: upper A prime at left parenthesis 4 comma 5 right parenthesis, upper B prime at left parenthesis 1 comma 3 right parenthesis, and upper C prime at left parenthesis 3 comma 1 right parenthesis. The coordinates of the plotted points for the third triangle upper A double prime upper B double prime upper C double prime joined by lines made of dashes and dots are as follows: upper A double prime at left parenthesis 1 comma 0 right parenthesis, upper B double prime at left parenthesis negative 2 comma negative 2 right parenthesis, and upper C double prime at left parenthesis 0 comma negative 4 right parenthesis.

How would you describe this series of transformations?

(1 point)
Responses

Translation of (2,0)
and then reflection across the x
-axis shows that triangle ABC
is congruent to triangle A′′B"C"
.
Translation of left parenthesis 2 comma 0 right parenthesis and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

Rotation of 90 degrees clockwise and then reflection across the x
-axis shows that triangle ABC
is congruent to triangle A′′B"C"
.
Rotation of 90 degrees clockwise and then reflection across the x -axis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

Reflection across the y
-axis and then a translation of (−3,−5)
shows that triangle ABC
is congruent to triangle A′′B"C"
.
Reflection across the y -axis and then a translation of left parenthesis negative 3 comma negative 5 right parenthesis shows that triangle upper A upper B upper C is congruent to triangle upper A double prime upper B double prime upper C double prime .

Since triangles ABC
and A′′B"C"
do not have the same orientation, they are not congruent.
Since triangles upper A upper B upper C and upper A double prime upper B double prime upper C double prime do not have the same orientation, they are not congruent.
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Answer 1

To analyze the transformations of the triangles based on their coordinates and determine congruence, we can look at the vertices of the triangles:

1. Triangle ABC has vertices:

   • A(-4, 5)
   • B(-1, 3)
   • C(-3, 1)

2. Triangle A'B'C' has vertices:

   • A'(4, 5)
   • B'(1, 3)
   • C'(3, 1)

3. Triangle A''B''C'' has vertices:

   • A''(1, 0)
   • B''(-2, -2)
   • C''(0, -4)

Steps to Identify the Transformations:

• Triangle ABC to Triangle A''B''C'':

  • To go from Triangle ABC to Triangle A''B''C'', we can check the transformations. Consider the point A(-4, 5) to A''(1, 0):
    • Translation: Moving (2, 0) gives A(-4 + 2, 5) = (-2, 5).
    • Then reflect across the x-axis: A''(1, 0) becomes (1, -5) which does not match.
  • Instead, we can observe the coordinates and deduce that a reflection across the y-axis and then a translation might better describe the transformation.

• Triangle A'B'C' to Triangle A''B''C'':

  • Here, consider going from A'(4, 5) to A''(1, 0):
    • A reflection and some adjustment in coordinates for translation will also come into play.

Conclusion:

Given the options:

1. Translation of (2,0) and then reflection across the x-axis shows that triangle ABC is congruent to triangle A''B''C''.
2. Translation of (2, 0) and then reflection across the x-axis shows that triangle A B C is congruent to triangle A double prime B double prime C double prime.

The transformations of Triangle ABC towards Triangle A'B'C' primarily indicate a reflection across the y-axis and then a translation downward. However, we noted a misalignment due to transformation descriptions.

From reviewing transformations, if the transformations maintain orientation considerations:

• Since triangles ABC and A''B''C'' do not have the same orientation, they are not congruent. (The option that indicates they are not congruent due to orientation differences would apply here.)

This means your correct choice should focus on orientation concerning congruency between triangles ABC and A''B''C''. The triangles show transformation through reflection or translation that alters their congruence based on final placements in the coordinate system.